The isomorphism problem for some universal operator algebras

TL;DR

This paper investigates the isomorphism problem for universal operator algebras generated by row contractions with polynomial relations, establishing conditions for isometric isomorphism and linking algebraic structures to geometric and analytic properties.

Contribution

It characterizes isometric isomorphisms of these algebras via polynomial relations and subproduct system isomorphisms, and connects algebraic geometry with operator algebra theory.

Findings

Two algebras are isometrically isomorphic iff their polynomial relations are unitarily equivalent.

Commutative operator algebras correspond to analytic functions on algebraic varieties.

A Nullstellensatz relates ideals to zero sets, aiding algebra classification.

Abstract

This paper addresses the isomorphism problem for the universal (nonself-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by radical relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C*-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the weak-operator closures of these algebras as well.